Stressability of Semi-Discrete Frameworks

TL;DR

This paper investigates the stressability of semi-discrete frameworks in the plane, characterizing it via difference-differential equations, and establishes a semi-discrete Maxwell-Cremona correspondence linking stressable frameworks to 3D conjugate surfaces.

Contribution

It introduces a new characterization of stressability for semi-discrete frameworks using difference-differential equations and extends the Maxwell-Cremona lifting property to the semi-discrete setting.

Findings

Stressability characterized by difference-differential equations.

Semi-discrete Maxwell-Cremona lifting property established.

Framework liftability linked to semi-discrete conjugate surfaces.

Abstract

In this paper we study the stressability of semi-discrete frameworks in the plane which are generated by a discrete sequence of smooth curves. We characterize their stressability property by the existence of stresses fulfilling certain difference-differential equation. In particular, we define a semi-discrete height function which we use to generate liftings. Furthermore, we show a semi-discrete analogue of the Maxwell-Cremona lifting property which implies that the stressable semi-discrete frameworks in the plane are precisely the orthogonal projections of semi-discrete conjugate surfaces in 3-space. Finally, we discuss geometric implications for frameworks with vanishing boundary forces and characterize the liftability of frameworks which only consist of two neighboring curves forming one strip.